D y n amic V irtual A rc C onsistency Hiep Nguyen 1 Christian - PowerPoint PPT Presentation

D y n amic V irtual A rc C onsistency Hiep Nguyen 1 Christian Bessiere 2 Thomas Schiex 1 1 INRA-BIA UR875, Toulouse, France 2 Universit´ e de Montpellier Montpellier, France JFPC’2013 H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 1 / 13

Problem Weighted Constraint Satisfaction Problem (WCSPs) Objectif Use Dynamic AC to speed up the Virtual Arc Consistency H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 2 / 13

Overview Weighted CSPs Virtual Arc Consistency Dynamic Virtual Arc Consistency Experiments Conclusion H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 3 / 13

Weighted CSPs X = { 1 , ..., n } variables D = { d 1 , ..., d n } domains H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 4 / 13

Weighted CSPs X = { 1 , ..., n } variables D = { d 1 , ..., d n } domains W = { w S 1 , ..., w S e } cost functions H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 4 / 13

Weighted CSPs X = { 1 , ..., n } variables D = { d 1 , ..., d n } domains W = { w S 1 , ..., w S e } cost functions H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 4 / 13

Weighted CSPs Minimimum cost assignment X = { 1 , ..., n } variables D = { d 1 , ..., d n } domains W = { w S 1 , ..., w S e } cost functions H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 4 / 13

Weighted CSPs Minimimum cost assignment X = { 1 , ..., n } variables D = { d 1 , ..., d n } domains W = { w S 1 , ..., w S e } cost functions ◮ w ∅ : 0-arity function that defines a LB on the cost of any solution. ◮ useful for Branch and Bound pruning ◮ significantly increased by Virtual Arc Consistency. H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 4 / 13

Virtual Arc Consistency (VAC − [AIJ2010]) Bool(P) Classic CSP induced by a WCSP P that authorizes only zero cost values and tuples (ignoring w ∅ ). H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 5 / 13

Virtual Arc Consistency (VAC − [AIJ2010]) 2 2 Bool(P) 1 Classic CSP induced by a WCSP P that authorizes 1 1 only zero cost values and tuples (ignoring w ∅ ). 1 P Bool( P ) H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 5 / 13

Virtual Arc Consistency (VAC − [AIJ2010]) 2 2 Bool(P) 1 Classic CSP induced by a WCSP P that authorizes 1 1 only zero cost values and tuples (ignoring w ∅ ). 1 VAC P P is VAC iff the AC closure of Bool( P ) is non-empty Bool( P ) H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 5 / 13

Virtual Arc Consistency (VAC − [AIJ2010]) 2 2 Bool(P) 1 Classic CSP induced by a WCSP P that authorizes 1 1 only zero cost values and tuples (ignoring w ∅ ). 1 VAC P P is VAC iff the AC closure of Bool( P ) is non-empty If P is not VAC: enforcing AC in Bool( P ) leads to a wipe out ∃ a way of shifting costs in P which leads to an increase of w ∅ . Bool( P ) H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 5 / 13

Enforcing VAC x 1 x 2 Iterative process a a enforcing AC in Bool( P ) until a wipe-out occurs b b transforming P into an equivalent problem with an increased w ∅ . a a Using “Equivalence Preserving Tranformations” to incrementally shift costs to the wipe-out variable. b b x 3 x 4 Bool( P ) Constraint revision order : w 13 , w 34 , w 12 , w 24 H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 6 / 13

Enforcing VAC x 1 x 2 Iterative process a a enforcing AC in Bool( P ) until a wipe-out occurs b b transforming P into an equivalent problem with an increased w ∅ . a a Using “Equivalence Preserving Tranformations” to incrementally shift costs to the wipe-out variable. b b x 3 x 4 Bool( P ) Constraint revision order : w 13 , w 34 , w 12 , w 24 H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 6 / 13

Enforcing VAC x 1 x 2 Iterative process a a 2 2 enforcing AC in Bool( P ) until a 1 wipe-out occurs b b transforming P into an equivalent 1 1 problem with an increased w ∅ . a a Using “Equivalence Preserving Tranformations” to incrementally shift costs to the wipe-out variable. 1 b b x 3 x 4 w ∅ = 0 H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 6 / 13

Enforcing VAC x 1 x 2 Iterative process a a 2 2 enforcing AC in Bool( P ) until a 1 wipe-out occurs b b transforming P into an equivalent 1 1 problem with an increased w ∅ . a a Using “Equivalence Preserving Tranformations” to incrementally shift costs to the wipe-out variable. 1 b b x 3 x 4 w ∅ = 0 H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 6 / 13

Enforcing VAC x 1 x 2 1 Iterative process a a 1 2 1 enforcing AC in Bool( P ) until a 1 wipe-out occurs b b transforming P into an equivalent 1 1 problem with an increased w ∅ . a a Using “Equivalence Preserving Tranformations” to incrementally shift costs to the wipe-out variable. 1 b b x 3 x 4 w ∅ = 0 H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 6 / 13

Enforcing VAC x 1 x 2 1 Iterative process a a 1 2 1 enforcing AC in Bool( P ) until a 1 wipe-out occurs b b transforming P into an equivalent 1 1 problem with an increased w ∅ . a a Using “Equivalence Preserving Tranformations” to incrementally shift costs to the wipe-out variable. 1 b b x 3 x 4 w ∅ = 0 H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 6 / 13

Enforcing VAC x 1 x 2 1 Iterative process a a 1 2 enforcing AC in Bool( P ) until a wipe-out occurs b 1 b transforming P into an equivalent 1 1 problem with an increased w ∅ . a a Using “Equivalence Preserving Tranformations” to incrementally shift costs to the wipe-out variable. 1 b b x 3 x 4 w ∅ = 0 H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 6 / 13

Enforcing VAC x 1 x 2 1 Iterative process a a 1 1 enforcing AC in Bool( P ) until a wipe-out occurs b b transforming P into an equivalent 1 1 problem with an increased w ∅ . a a Using “Equivalence Preserving Tranformations” to incrementally shift costs to the wipe-out variable. 1 b b x 3 x 4 w ∅ = 1 H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 6 / 13

Motivation for Dynamic VAC VAC enforces AC on a sequence of incrementally modified CNs Maintaining Bool( P ) by Dynamic AC ⇒ Dynamic VAC H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 7 / 13

Dynamic VAC Property Each VAC iteration leads only to constraint relaxations in Bool ( P ) Maintaining Bool( P ) by dynamic AC (AC / DC2 − [FLAIRS 2005] ) Relaxation proceduce: Restoring restorable values (see above) 1 Propagating restored values to neighborhood (new support) 2 Rechecking the restored values for AC 3 H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 8 / 13

Avantages of DynVAC . Bool(P) Saving work by keeping viable values and some deleted values from the previous iterations H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 9 / 13

Maintaining VAC in search Going down x � = a , x ≤ a , x ≥ a : domain restriction, done by AC x = a : if ( x , a ) has a positive cost ◮ this cost goes directly to w ∅ ◮ relaxation of Bool ( P ) : done by relaxation proceduce Going up rebuilding all domains in Bool(P) thanks to the ”killer” data-structure (backtrackable) H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 10 / 13

Experimentation: pre-processing Cost Function Library DynVAC is faster than VAC for celar (1.6x), tagsnp (3x), warehouse (5x), but significantly slower for maxclique problems. A domain based heuristic handles those pathological cases. H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 11 / 13

VAC DynVAC ratio time(sec) time(sec) time nodes cl6-2 29 46 1,59 1.17 gr11 470 366 0,78 0,65 CELAR gr13 1.431 1.144 0,8 0,51 sc06-18 1.511 736 0,49 0,77 Results in search sc06-20 765 508 0,66 1,26 sc06 5.227 2.454 0,47 0,79 ... capa 2.462 1.013 0,41 1,00 WAREHOUSE capb 3.019 6.168 2,04 1,35 capc 2.027 1.228 0,61 0,75 capmo5 75 14 0,19 0.74 capmq1 5.111 2.374 0,46 0,68 capmq2 6.520 3.209 0,49 1,05 ... average (47 prob) 1831 1117 0,61 0,83 Only some worse and best cases are presented. H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 12 / 13

Conclusion DynVAC incremental algorithm for enforcing VAC in WCSPs faster than VAC for large costs and large domains problems a heuristic that gets rid of pathological cases Perspective extension to non-binary cost functions H. Nguyen et al. (INRA and LIRMM) D y n amic V irtual A rc C onsistency JFPC’2013 13 / 13